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Assessing the impact of the realized range on the (E)GARCH volatility: Evidence from Brazil Resumo O presente artigo investiga se a inclusão do realized range como regressor na equação da volatilidade (E)GARCH adiciona informação ao processo, melhorando a performance das previsões e fornecendo estimativas mais precisas para a persistência da volatilidade. Dezesseis definições do realized range, em onze frequências amostrais, são testadas utilizando dados do mercado brasileiro. Diversos critérios foram utilizados para avaliar se ocorreram melhorias nos ajustes, incluindo o teste da razão das verossi- milhanças, a redução da persistência e um teste estatístico para comparar erros de previsão de modelos concorrentes. Descobrimos que sempre há, para ambos os modelos GARCH e EGARCH, algum tipo de realized range que, em diversas frequências, explica melhor o processo da volatilidade acarretando uma redução considerável da persistência. Palavras-chaves: modelos GARCH e EGARCH; Volatilidade Realizada; Realized range; Previsão de volatilidade. Abstract This paper investigates whether the inclusion of the realized range as regressor in the (E)GARCH volatility equation would add information to the process improving out-of-sample forecasts performance and providing more accurate estimates of the volatility persistence. Sixteen range measures at eleven data frequencies are tested using Brazilian stock market data. Several measures for assessing the im- provements in the fits were used including the likelihood ratio test, the persistence percentage decrease, and a formal statistical test for comparing forecasts errors from competing models. We found that for both the GARCH and EGARCH models there are always some realized range type at some frequencies bringing information to the volatility process with considerable persistence reduction. Key words: GARCH and EGARCH models; Realized volatility; Realized range; Volatility forecast. JEL Classification: C22 C58 C32 1

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Page 1: Assessing the impact of the realized range on the (E)GARCH ... · Assessing the impact of the realized range on the (E)GARCH volatility: Evidence from Brazil Resumo O presente artigo

Assessing the impact of the realized range on the (E)GARCH volatility:Evidence from Brazil

Resumo

O presente artigo investiga se a inclusão do realized range como regressor na equação da volatilidade

(E)GARCH adiciona informação ao processo, melhorando a performance das previsões e fornecendo

estimativas mais precisas para a persistência da volatilidade. Dezesseis definições do realized range,

em onze frequências amostrais, são testadas utilizando dados do mercado brasileiro. Diversos critérios

foram utilizados para avaliar se ocorreram melhorias nos ajustes, incluindo o teste da razão das verossi-

milhanças, a redução da persistência e um teste estatístico para comparar erros de previsão de modelos

concorrentes. Descobrimos que sempre há, para ambos os modelos GARCH e EGARCH, algum tipo de

realized range que, em diversas frequências, explica melhor o processo da volatilidade acarretando uma

redução considerável da persistência.

Palavras-chaves: modelos GARCH e EGARCH; Volatilidade Realizada; Realized range; Previsão de

volatilidade.

Abstract

This paper investigates whether the inclusion of the realized range as regressor in the (E)GARCH

volatility equation would add information to the process improving out-of-sample forecasts performance

and providing more accurate estimates of the volatility persistence. Sixteen range measures at eleven

data frequencies are tested using Brazilian stock market data. Several measures for assessing the im-

provements in the fits were used including the likelihood ratio test, the persistence percentage decrease,

and a formal statistical test for comparing forecasts errors from competing models. We found that for

both the GARCH and EGARCH models there are always some realized range type at some frequencies

bringing information to the volatility process with considerable persistence reduction.

Key words: GARCH and EGARCH models; Realized volatility; Realized range; Volatility forecast.

JEL Classification: C22 C58 C32

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Assessing the impact of the realized range on the (E)GARCH volatility:Evidence from Brazil

1 Introduction

Volatility is a key piece in the financial environment, with significant role in investment, securityvaluation, risk management, and monetary policy making. Usually most of the activity in theresearch area of a financial institution is devoted to the modeling and forecasting of an assetvolatility.

Among all different volatility modeling approaches, the GARCH family is unequivocallythe most popular and well known one, probably due to its flexibility, easy to estimate andsuccess when modeling many of the stylized facts about financial returns, in particular thepresence of clusters of volatility. For a comprehensive review see Poon and Granger (2003).Since Engle’s (1982) seminal paper, a long list of articles have empirically proved the usefulnessof the GARCH family members in areas such as economics and social sciences (see, e.g., theworks of Bollerslev (1986, 1987), Engle et al. (1987), Nelson (1991), Glosten et al. (1993),and Hansen and Lunde (2005a)). However, their out-of-sample predictive ability was cast indoubt after some unsatisfactory empirical results, see Akgiray (1989), Kat and Heynen (1994),Franses and Van Dijk (1995), Brailsford and Faff (1996) and Figlewski (1997)). It should benoted, though, that the poor out-of-sample performance could be accredited to the inherentnoise of the squared returns used in many works as proxies in the evaluation criteria. In fact,Andersen and Bollerslev (1998) using as proxy a new measure based on high-frequency returnswere able to show that GARCH volatility daily forecasts do perform well.

To further improve GARCH volatility forecasts one may include exogenous variables in theconditional variance equation. Usual choices of regressors include trading volume, macroe-conomic news announcements, overnight returns, after hours volatility, implied volatility fromoption prices and realized volatility.

However, there is no consensus on the advantages of the use of such exogenous variables.For example, with respect to the addition of trading volume, Lamoureux and Lastrapes (1990)and Brooks (1998) found conflicting results. Using daily equity returns and GARCH mod-els, Flannery and Protopapadakis (2002) investigated seventeen macroeconomic variables andfound that six of them (Consumer Prices Index, Producer Price Index, a Monetary Aggregate,Balance of Trade, Employment Report, and Housing Starts) could be strong risk factor candi-dates. Similarly, Bomfim (2003) found that macroeconomic pre-announcement effects in the

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stock market could be highly significant.Gallo and Pacini (1998), using six stock market indices, found evidence of improvements

in the forecasting performance of the GARCH and EGARCH models when the overnight re-turns are inserted into the conditional variance equation. Martens (2002) showed that for theS&P 500 returns, the squared overnight return plus the realized volatility as well as the rescaledrealized volatility are statistically significant. Taylor (2007) also found evidence of significantimprovements in the conditional volatility forecasts of the S&P 500 index returns when includ-ing overnight information revealed in its E-mini futures market. Chen et al. (2012), by breakingup the after-hours period into pre-open, post-close and no-trading, investigate the role of thepre-open period variance in the GARCH conditional volatility.

The implied volatility seems also to bring important additional information to the condi-tional variance equation of some series. Blair et al. (2001) found that the index VIX impliedvolatility accounted for almost all relevant information about the conditional variance of theS&P 100 index daily returns. Giot (2003) and Koopman et al. (2005) showed the superiorityof implied volatility for options, see also Day and Lewis (1992). We should note that even inmature markets not every stock has an option being traded. Hence, for these cases, model freerealized measures of volatility may provide reliable exogenous alternatives. For example, Tay-lor and Xu (1997) included the 5-minute realized volatility and the daily implied volatility con-structed based on exchange rate quotations to explain the dynamics of the Deutschemark/dolarvolatility. They found that the former was not quite helpful for daily estimation, although ap-pearing to be more informative than the later when hourly data was used. Zhang and Hu (2013)experimented the realized volatility as a regressor but did not provide a final answer to the ques-tion whether the realized measure could actually provide additional information to the volatilityprocess. Thus, there is more to be explored in the world of the realized measures and theiradditional information to GARCH models.

Accordingly, in this paper we use Brazilian data to investigate whether the inclusion ofthe realized range as exogenous variables in the variance equation of GARCH and EGARCHmodels can provide additional information to the volatility process improving the one-step-ahead forecasts and providing more accurate estimates of the persistence. Our methodology issimilar to those found in Zhang and Hu (2013) and Day and Lewis (1992), differing from thatwhen we include the log-realized range as regressor, and also when we use formal statisticaltests and graphics to evaluate the performance of the augmented model.

Our decision in considering Brazilian data is due to two reasons. First, the inexpressivenumber of investigations on volatility models based on high-frequency data devoted to emerg-ing markets. Second, the still open question whether or not an emerging market may provide

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valuable insights for foreign investors interested in investing in emerging or even in developedmarkets. Although Merton (1987) suggests that investors holding imperfectly diversified portfo-lios (those that cannot be fully diversify away firm-specific risk due to limited access to informa-tion in the presence of market frictions), and some economic theories suggest that idiosyncraticvolatility should be positively related to expected returns (Xu and Malkiel (2003) and Ewenset al. (2013)), Ang et al. (2006, 2009) found a negative relation between idiosyncratic volatilityand future returns. Measuring the stock return idiosyncratic volatility relative to the Fama andFrench (1993) model, Ang et al. (2006, 2009) investigated 23 developed markets, with a closeranalysis of the U.S. market for which more data were available, concluding that the puzzle oflow returns to high idiosyncratic volatility stocks calls for further investigation. Choudhry andOsoble (2015) highlight that although the investigation of interdependence amongst interna-tional stock markets has important practical implications for asset allocation, risk management,and economic policy, the existing research on industry or firm level interdependence has beenmainly focusing on developed markets.

The contribution of this paper is twofold. Firstly, we empirically show through tests andgraphics that the realized range may provide further information to GARCH and EGARCH pro-cesses, resulting in better forecasts and persistence reduction. Secondly, although only Braziliandata is analyzed, this study may provide valuable insights for foreign investors interested in in-vesting in emerging markets.

The remainder of this paper is organized as follows. The following two sections provide thetheoretical framework for the empirical analysis, including a brief discussion on high-frequencydata, the definitions of the realized measures, and a brief review of the (augmented) GARCHclass of models. Section 4 describes the Brazilian stock market and the data sets used, carry onthe analysis and discusses the empirical findings. Concluding remarks are given in Section 5.

2 Realized Measures

High-frequency data have recently become widely available, promoting the development offaster and more sophisticated computer resources requiring from researchers beyond basic pro-gramming skills. In spite of being very expensive, some of their sub-products, such as therealized measures, indexes and proxies, have become very popular. Andersen et al. (1999)advocated that the rich information contained in the high-frequency data may be effectivelyexploited through the use of realized volatility measures.

According to varying assumptions made on information sets and frequencies, different re-alized volatility measures may be extracted from high frequency data. Andersen et al. (1999)

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showed that the realized volatilities provide a more meaningful and accurate alternative whenevaluating the performance of daily or longer-run volatility forecasts. In fact, the two moreimportant applications for these observed series may be (i) their ability for improving or moreaccurately assess forecasts performance, and (ii) helping quantifying market risk, see Andersenand Bollerslev (1998), Engle (2000), Barndorff-Nielsen and Shephard (2002b), Andersen et al.(2011), among others. In this paper we explore several definitions of realized measures andtheir usefulness in these two directions.

2.1 Realized Volatility

The term realized volatility was first used by Fung and Hsieh (1991) and later on by Andersenand Bollerslev (1998) to denote the sum of intraday squared returns computed at short intervals.The idea was to propose a new estimator for the unobserved integrated variance over timeintervals. As such, it was intended to be used as an indicative of future price movements. It isa model free, computationally trivial and, in principle, highly accurate estimator, see Dieboldand Strasser (2012).

Under the Andersen et al. (2001b) framework, the logarithmic asset price (p) incrementsevolve continuously through time according to a stochastic volatility diffusion process, and thequadratic variation theory guarantees that the sum of the intraday squared returns converges tothe integrated variance. The realized variance estimator (RV ) over a time interval t of 1 day isdefined as

RVt(∆) =n

∑j=1

r2t−1+ j∆, (1)

where ∆ = 1/n is the sampling interval, n is the number of intervals within 1 day, and rt−1+ j∆ =

pt−1+ j∆− pt−1+( j−1)∆ defines the intraday ∆-frequency continuously compounded return.Assuming the absence of jumps and microstructure noise, the ex-post realized variance is an

unbiased and consistent estimator of the integrated variance (IV) when the sampling frequencytheoretically goes to the continuous basis. In his seminal paper, Merton (1980) had alreadyobserved that, in the theoretical limit of continuous observation, the variance could in principlebe estimated without error for any finite interval.

Therefore, in practice, the optimality of the realized variance as an estimate of the dailyintegrated variance is attached to smaller sampling intervals ∆, which could be as small as thoseprovided by the transaction-to-transaction prices. This sampling issue was addressed in severalstudies, see, e.g., Andersen et al. (2001a,b, 2003), Barndorff-Nielsen and Shephard (2002a,b),where sampling intervals of five to thirty minutes were considered. Bandi and Russell (2008)derived a mean squared error based optimal sampling frequency resulting in more accurate

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forecasts and some economic gains. Some recent works (see, e.g., Zhang et al. (2005), Aït-Sahalia et al. (2005), Hansen and Lunde (2006), Aït-Sahalia et al. (2011), and Barndorff-Nielsenet al. (2008)) have used all available intraday information, giving rise to the so called robust

realized variance estimators.

2.2 Realized Range

A drawback of the realized variance as an estimator of the integrated variance is its strongdependence on the sampling interval. This issue may be better handled by the realized range,whose theoretical properties were derived in Martens and van Dijk (2007) and Christensen andPodolskij (2007).

Parkinson (1980) derived an estimator for the price range under the assumption that thedaily asset price follows a simple diffusion model without a drift term. Noting that modelsbased only on closing prices ignore the prices inside the reference period, the definition waslater on extended to any interval, in particular to intraday intervals. The Parkinson realizedrange estimator RRParkinson

t (∆) is defined as

RRParkinsont (∆) =

14ln2

1/∆

∑j=1

(u j−d j)2 (2)

u j = lnH j− lnO j d j = lnL j− lnO j (3)

where H j, L j and O j are the highest, the lowest and the opening prices of the jth interval in thetth trading day; and u j and d j are normalized high and low prices.

Other definitions of realized ranges were then proposed. Extending the definition of thevolatility estimator of Garman and Klass (1980), Martens and van Dijk (2007) proposed theRRGK1

t (∆) and the RRGK2t (∆) estimators:

RRGK1t (∆) =

1/∆

∑j=1

[0.511(u j−d j)2−0.019[c j(u j +d j)−2u jd j]−0.383c2

j ] (4)

RRGK2t (∆) =

1/∆

∑j=1

[0.5(u j−d j)2− (2ln2−1)c2

j ] (5)

c j = lnC j− lnO j (6)

where c j is a normalized closing price based on C j, the closing price of the jth interval onthe tth trading day, and where u j, d j and O j are as previously defined. Through simulation

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experiments, these range estimates were found to be more efficient than (2).Assuming a non-zero drift term, Rogers and Satchell (1991) proposed the following range

estimator

RRRSt (∆) =

1/∆

∑j=1

[u j(u j− c j)+d j(d j− c j)], (7)

where u j, d j and c j are as previously defined. The authors proved that RRRSt (∆) is an unbiased

volatility estimator whatever is the drift.

2.3 Construction of the Realized Measures

In order to construct the realized measures used in the empirical investigation of Section 4, wediscuss two important issues.

The first one is concerned with the sampling frequency ∆. Noting that several studies foundthe optimal choice for the sampling frequency ∆ to be between one and thirty minutes (see,e.g., Andersen et al. (2001a, 2003), Hansen and Lunde (2006) and Bandi and Russell (2008)),we take the following choices for ∆ in minutes: {1/60,1,5,10,15,20,30,60,105,210,420}.Since the trading frequency and volume are smaller during the after-market session, when itis observed only up to 2% price variation relative to the regular session closing price, the ∆-frequency intervals are computed within the regular session hours only. For example, using∆ = 15 in a 420 minutes normal trading day results in n = 28 observations of the realizedmeasure.

The second problem is how to extend the definition of a realized measure so that it representsa volatility measure for the whole day. We note that the earliest studies on realized volatilitywere based on exchange rate data, which are available 24 hours/day (apart from weekends).However, in the case of financial instruments, usually traded only in the active part of the day,one should somehow include the overnight data. Likewise Hansen and Lunde (2005b), weconsider three ways for extending the realized measures of volatility: scaling the estimatoror taking the squared overnight returns and either adding to (or combining with) the (daily)realized measure.

Let Rt(∆) represent any previously defined realized range (2), (4), (5) or (7), and let r1,t

denote the overnight return, that is, the close-to-open return, on date t. The first proposal scalesRt(∆) by a constant so that the resulting estimator has the desired expected value.

Rscalet (∆) = δRt(∆), (8)

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where δ may be estimated (Hansen and Lunde, 2005b) using

δ̂ =T

∑t=1

(rt− r̄)2/T

∑t=1

Rt , (9)

where T is the sample size.The second proposal just adds the squared overnight return

R+ont (∆) = r2

1,t +Rt(∆). (10)

Noting that both Rscalet (∆) and R+on

t (∆) are actually linear combinations of r21,t and Rt(∆) with

weights (0,δ ), and (1,1) a more general extension may be obtained as

Rωt (∆) = ω1r2

1,t +ω2Rt(∆), (11)

where the optimal weights ω1 and ω2 are those values minimizing the Mean Squared Error(MSE). Hansen and Lunde (2005b) derived the expression for the weights (ω1,ω2)

ω1 = (1−ϕ)µ0

µ1ω2 = ϕ

µ0

µ2(12)

where µ0 = E[IV], µ1 = E[r21,t ], µ2 = E[Rt ], ϕ =

µ22 η2

1−µ1µ2η12µ2

2 η21+µ2

1 η22−2µ1µ2η12

is a relative importance

factor, where η21 = var(r2

1,t), η22 = var(Rt), and η12 = cov(r2

1,t ,Rt). The authors also proposedthe sample estimates µ̂0 = (1/T )∑

Tt=1(r

21,t + Rt), µ̂1 = (1/T )∑

Tt=1 r2

1,t , µ̂2 = (1/T )∑Tt=1 Rt ,

η̂21 = (1/T )∑

Tt=1(r

21,t − µ̂1)

2, η̂22 = (1/T )∑

Tt=1(Rt − µ̂2)

2, and η̂12 = (1/T )∑Tt=1 Rt(r2

1,t − µ̂1),where T is the sample size.

3 The Augmented GARCH Conditional Volatility

Volatility is not observable and clearly time varying. It shows periods of high volatility andperiods of apparent calm, resulting in the well known clustering stylized fact. The Autoregres-sive Conditionally Heteroskedastic (ARCH) process of Engle (1982) successfully captures thevolatility clusters and generalizes the implausible assumption of constant variance forecast. Thebasic idea behind this model is that the asset returns innovations are serially uncorrelated witha conditional variance which is a function of squared lagged returns. A generalized version ofthis process, the Generalized Autoregressive Conditionally Heteroskedastic (GARCH) process,with a more flexible lag structure was proposed by Bollerslev (1986).

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Let Pt be the asset price at day t and rt = ln(Pt)− ln(Pt−1) be the continuously compoundedreturn on the asset over the period t−1 to t. Let rt = µt +

√htεt , where µt = E[rt | Ψt−1] and

ht = var(rt |Ψt−1) are respectively the conditional mean and variance given the information setΨt−1 at time t−1, and εt ∼ i.i.d.(0,1). The GARCH(p,q) conditional variance is given by

ht = α0 +p

∑i=1

αir2t−i +

q

∑j=1

β jht− j, (13)

where αi and β j are nonnegative parameters with ∑pi=1 αi +∑

qj=1 β j < 1 and α0 > 0. The un-

conditional variance of rt is given by α0

(1−∑

pi=1 αi−∑

qj=1 β j

)−1.

Motivated by the GARCH model limitation of responding equally to positive and negativeinnovations, Nelson (1991) proposed a model for the logarithm of the conditional variancewhich also captures the effect of bad news. The Exponential GARCH (EGARCH) processspecifies that

lnht = α0 +p

∑i=1

αi(| rt−i |√

ht−i−E[| rt−i |√

ht−i])+

q

∑j=1

β j lnht− j +p

∑i=1

γirt−i√ht−i

. (14)

It is worthy noting that taking the logarithm of the conditional variance simplifies greatly theestimation procedure.

We now include the realized range as exogenous variable in the volatility equations (13) and(14). From now on we assume µt = 0 or that it has been estimated and subtracted from thereturn series. To unify the notations given in Section 2.3, let R∗t (∆) represent any of the Rt(∆)

measures, or their three extensions, Rscalet (∆), R+on

t (∆), and Rωt (∆), or their logarithms. The

augmented GARCH and EGARCH conditional variance equations estimated in the empiricalinvestigation in Section 4 are:

ht = α0 +α1r2t−1 +β1ht−1 +θ1R∗t−1(∆), (15)

and

lnht = α0 +α1(| rt−1 |√

ht−1−E[| rt−1 |√

ht−1])+β1 lnht−1 + γ1

rt−1√ht−1

+θ1R∗t−1(∆). (16)

In (15) the explanatory variable and its coefficient θ1 must be positive, whereas in (16) thereare no restrictions. Therefore, in the applications that follow, we consider the logarithm of therealized measures only in (16). We observe that a more general model with several laggedvalues of the regressor may be considered.

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Equation (15) may be rewritten as

ht = α0 +α1(r2t−1−ht−1)+(α1 +β1)ht−1 +θ1R∗t−1(∆). (17)

The term (r2t−1−ht−1) in (17) is known as the shock to volatility and possesses zero expected

value. Thus, in the long run, the conditional volatility value has two important components.One is the previous value times (α1 +β1), and the other the positive quantity brought in by theexplanatory variable. Clearly, (α1 +β1) represents the persistence of the volatility, the rate atwhich the conditional variance reverts to its unconditional value. For the EGARCH model thepersistence equals to β1.

Models are estimated by maximum likelihood using the R environment and the rugarch

package (Ghalanos, 2013). We test the explanatory power of the regressor using a t-test andassess the statistical significance of its contribution applying the standard likelihood ratio test(LRT). To test whether or not the inclusion of the realized range in the volatility equation im-proves forecasts and reduces persistence we apply formal tests and graphics. These evaluationprocedures will be explained in the next section.

4 Data and Empirical Analysis

The dataset was obtained directly from BM&FBOVESPA, which trading scheme is consistedof two trading sessions, regular session and after-market, both preceded by pre-opening auctionsessions. The continuous regular session, in Brasília time (BRT) zone, begins at 10:00 andends at 17:00 with its pre-opening session from 9:45 to 10:00. The after-market is from 17:35until 19:00 with its pre-opening session from 17:30 to 17:35. The trading hours may changeaccording to the daylight saving program, with the addition of one hour.

The data are composed by the tick-by-tick transactions information on time, price and vol-ume, from December 1st 2009 to March 23rd 2012, on the eight currently most liquid con-stituent stocks of the Bovespa Index, namely the PETR4 (Petrobras), VALE5 (Vale), TNLP4(Telemar), USIM5 (Usiminas), BBDC4 (Banco Bradesco), CSNA3 (Companhia SiderúrgicaNacional), OGXP3 (OGX Petróleo) and ITUB4 (Itauunibanco). Table 7 in the Appendix gath-ers some summary statistics on the corresponding daily log-returns series of 572 observations.In this table the stocks are ordered according to their standard deviations. Columns 7 and 8give the values of the asymmetry and excess kurtosis sample coefficients, where the ∗ indicateswhich returns distributions are asymmetric with heavier tails than the Gaussian distribution atthe 5% level. The larger excess kurtosis were observed for OGXP3 and TNLP4. The Jarque-

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Bera test for normality rejected the null hypothesis for all stocks.The 20 realized measures (3 extensions on 5 definitions) are then computed for each ∆

frequency (a total of 11 ∆ values) using the high frequency data for the 8 stocks. We recallthat the main features of the unconditional distribution of stocks realized volatility are muchless known for emerging markets. Regarding the realized range, the number of studies are evenmore limited. We illustrate and give in tables 1 and 2, for all stocks and for ∆ = 5-minutes,some summary statistics of the unconditional distribution of the logarithm of the RRParkinson

t (∆)

and its three extensions, and the logarithm of the RV t(∆), respectively.In tables 1 and 2 we observe that all measures’ distributions present large positive skewness

and excess kurtosis, a well known stylized fact about volatility measures. The R+ont definition

produces, in general, the smallest minimum and the largest maximum, standard deviation andskewness coefficient. This behavior is in line with others previous works which credit this largervariability to the equal weights given to the daily and overnight quantities.

4.1 Estimating the augmenting volatility equation

After applying the Ljung-Box test to the daily squared returns, and the Lagrange multiplier testfor ARCH effects on the daily returns, and rejecting their null hypotheses for all stocks at the5% significance level, we proceed in the estimation of the GARCH-type models.

We estimate by maximum likelihood the parameters of the GARCH and EGARCH specifi-cations (13) and (14) with p= q= 1 (the search for the best orders for the large majority of casesindicated the simple GARCH(1,1) as the best model). All solutions provided by the statisti-cal computer package were carefully checked following suggestions given in McCullough andVinod (2003). Then, all realized range measures (32 types and 11 frequencies) were includedin the volatility equations, and (15) and (16) were estimated. We recall that for the GARCHmodel the log-realized measures were not considered as exogenous variables.

The GARCH(1,1) and EGARCH(1,1) models based on either the Normal or on the t-student conditional distribution resulted in excellent fits for all stocks, in the sense that theywere able to capture the dynamics observed in the second moment of the returns distributions.Using a fatter tail asymmetric conditional distribution did not further improve the fits. Theaugmented models were then estimated keeping the winning specification.

We now look for indicators of whether or not the realized range helps explaining the evo-lution of the conditional (E)GARCH variance. Typically, for any given stock, all tested re-alized range measures R∗t (∆) for some frequencies ∆ (we experimented 176 specifications inthe case of the GARCH model, and 352 in the case of the EGARCH model) were found sta-tistically significant improving the fit as measured by the likelihood ratio test. There were no

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Table 1: Summary statistics of the logarithm of the Parkinson realized range (∆ = 5 minutes)and its three extensions.

Asset R∗t Mean Min Max StdDev Skew Kurt ω1 ω2

BBDC4 lnRt 0.6748 -0.9166 2.7518 0.5575 0.7330 1.3116 0.0000 1.0000lnRscale

t 0.8381 -0.7533 2.9151 0.5575 0.7330 1.3116 0.0000 1.1774lnR+on

t 0.9556 -0.6516 3.4623 0.6401 0.8348 1.2195 1.0000 1.0000lnRω

t 1.0228 -0.5726 3.0810 0.5580 0.7407 1.2818 0.0653 1.3895VALE5 lnRt 0.3010 -1.2660 3.0443 0.5657 0.6296 1.4521 0.0000 1.0000

lnRscalet 0.8897 -0.6773 3.6330 0.5657 0.6296 1.4521 0.0000 1.8016

lnR+ont 0.7197 -1.2090 3.7136 0.7326 0.8626 1.3426 1.0000 1.0000

lnRωt 0.8702 -0.7311 3.5769 0.5710 0.6332 1.3528 0.0912 1.7019

ITUB4 lnRt 0.6636 -1.0988 2.9103 0.5575 0.8249 1.4918 0.0000 1.0000lnRscale

t 0.9185 -0.8440 3.1652 0.5575 0.8249 1.4918 0.0000 1.2903lnR+on

t 0.9527 -0.7906 3.7341 0.6752 1.0293 1.9009 1.0000 1.0000lnRω

t 1.0654 -0.6934 3.3106 0.5578 0.8248 1.4860 0.0069 1.4914PETR4 lnRt 0.5959 -0.8919 2.7741 0.5235 0.5951 1.3263 0.0000 1.0000

lnRscalet 1.0013 -0.4865 3.1795 0.5235 0.5951 1.3263 0.0000 1.4999

lnR+ont 0.9143 -0.7892 3.5600 0.6483 0.8287 1.2875 1.0000 1.0000

lnRωt 1.0149 -0.4845 3.1827 0.5254 0.5931 1.2787 0.0454 1.4980

TNLP4 lnRt 0.8740 -0.5705 2.6628 0.5482 0.4266 0.3129 0.0000 1.0000lnRscale

t 1.1589 -0.2856 2.9476 0.5482 0.4266 0.3129 0.0000 1.3296lnR+on

t 1.0955 -0.4776 3.7518 0.6221 0.8475 1.5632 1.0000 1.0000lnRω

t 1.1732 -0.2698 3.0025 0.5477 0.3563 0.2582 -0.1785 1.4074CSNA3 lnRt 1.0657 -1.2157 3.2014 0.5783 0.4962 1.0713 0.0000 1.0000

lnRscalet 1.2730 -1.0084 3.4087 0.5783 0.4962 1.0713 0.0000 1.2303

lnR+ont 1.3591 -1.1102 4.1433 0.6811 0.7853 1.3278 1.0000 1.0000

lnRωt 1.4552 -0.8217 3.5968 0.5783 0.4904 1.0704 -0.0189 1.4849

USIM5 lnRt 1.1869 -0.9518 3.2429 0.5840 0.3590 0.8030 0.0000 1.0000lnRscale

t 1.5026 -0.6360 3.5587 0.5840 0.3590 0.8030 0.0000 1.3713lnR+on

t 1.4933 -0.9474 4.0022 0.6516 0.5590 1.1916 1.0000 1.0000lnRω

t 1.5551 -0.6180 3.5821 0.5807 0.3807 0.8302 0.1065 1.3959OGXP3 lnRt 1.3365 0.1080 3.9453 0.5850 0.8484 1.4516 0.0000 1.0000

lnRscalet 1.7599 0.5314 4.3687 0.5850 0.8484 1.4516 0.0000 1.5271

lnR+ont 1.6615 0.1808 5.7882 0.7137 1.0218 2.5203 1.0000 1.0000

lnRωt 1.8201 0.6035 4.4514 0.5822 0.8393 1.4192 -0.0668 1.6588

Summary statistics of the logarithm of the Parkinson realized range and its three extensions across the 8 stocks. Rscalet , R+on

t , and Rt(ω) refer to

definitions (8), (10) and (11), respectively. Kurt stands for the excess kurtosis, and ω1 and ω2 are the optimal weights. The sampling frequency

is ∆ = 5 minutes.

differences between the GARCH and the EGARCH models with respect to how many realizedmeasures specifications were statistically significant. For both models the winning frequenciesat which the realized ranges brought in exploratory power are, ordered by number of cases,1-second, 210-minutes, 420-minutes, 105-minutes and 30-minutes. The Rogers and Satchell

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Table 2: Summary statistics of the logarithm of the realized variance (∆ = 5 minutes) and itsthree extensions.

Asset R∗t Mean Min Max StdDev Skew Kurt ω1 ω2

BBDC4 lnRt 0.6268 -0.9036 2.9614 0.6054 0.6486 0.8816 0.0000 1.0000lnRscale

t 0.8079 -0.7225 3.1425 0.6054 0.6486 0.8816 0.0000 1.1985lnR+on

t 0.9262 -0.6641 3.4068 0.6697 0.7096 0.8933 1.0000 1.0000lnRω

t 0.9812 -0.5724 3.2530 0.6045 0.6666 0.8258 0.2056 1.3370VALE5 lnRt 0.2220 -1.5360 3.1365 0.6443 0.3861 0.8440 0.0000 1.0000

lnRscalet 0.8438 -0.9143 3.7582 0.6443 0.3861 0.8440 0.0000 1.8622

lnR+ont 0.6684 -1.3683 3.7133 0.7816 0.6709 1.0139 1.0000 1.0000

lnRωt 0.8062 -0.9527 3.6539 0.6468 0.4267 0.7929 0.1538 1.6755

ITUB4 lnRt 0.6260 -1.0956 3.1499 0.6143 0.6731 1.0466 0.0000 1.0000lnRscale

t 0.8827 -0.8389 3.4066 0.6143 0.6731 1.0466 0.0000 1.2927lnR+on

t 0.9303 -1.0878 3.7513 0.7095 0.8669 1.4999 1.0000 1.0000lnRω

t 1.0273 -0.7219 3.5235 0.6160 0.6853 1.0136 0.0871 1.4525PETR4 lnRt 0.4941 -1.1343 2.9070 0.6092 0.4376 0.8731 0.0000 1.0000

lnRscalet 0.9494 -0.6789 3.3623 0.6092 0.4376 0.8731 0.0000 1.5768

lnR+ont 0.8442 -1.0051 3.4904 0.7088 0.6014 0.8301 1.0000 1.0000

lnRωt 0.9299 -0.7490 3.2912 0.6126 0.4422 0.7509 0.1908 1.4438

TNLP4 lnRt 1.0385 -1.5381 2.7331 0.5617 0.1083 0.5038 0.0000 1.0000lnRscale

t 1.1596 -1.4170 2.8541 0.5617 0.1083 0.5038 0.0000 1.1287lnR+on

t 1.2380 -0.6182 3.7592 0.6177 0.5945 0.9228 1.0000 1.0000lnRω

t 1.2961 -1.3890 3.0000 0.5620 0.0838 0.6210 -0.0481 1.3076CSNA3 lnRt 1.1711 -1.2944 3.3629 0.5854 0.3345 0.8945 0.0000 1.0000

lnRscalet 1.2742 -1.1913 3.4660 0.5854 0.3345 0.8945 0.0000 1.1086

lnR+ont 1.4489 -1.1807 4.1167 0.6686 0.6678 1.2335 1.0000 1.0000

lnRωt 1.5281 -0.9455 3.7088 0.5845 0.3485 0.9047 0.0367 1.4131

USIM5 lnRt 1.2014 -0.6665 3.4633 0.6068 0.2882 0.6738 0.0000 1.0000lnRscale

t 1.4901 -0.3779 3.7520 0.6068 0.2882 0.6738 0.0000 1.3346lnR+on

t 1.5127 -0.5686 4.0156 0.6570 0.4835 0.9581 1.0000 1.0000lnRω

t 1.5645 -0.2731 3.7642 0.5977 0.3376 0.6717 0.1947 1.3472OGXP3 lnRt 1.2551 -0.3509 3.9702 0.6417 0.6759 1.0293 0.0000 1.0000

lnRscalet 1.7232 0.1172 4.4383 0.6417 0.6759 1.0293 0.0000 1.5969

lnR+ont 1.6055 -0.1914 5.7936 0.7502 0.8838 2.1122 1.0000 1.0000

lnRωt 1.7540 0.1660 4.4937 0.6429 0.6508 0.9602 -0.0653 1.6880

Summary statistics of the three extensions of the realized volatility across the 8 stocks. Rscalet , R+on

t , and Rt(ω) refer to definitions (8), (10) and

(11), respectively. Kurt stands for the excess kurtosis, and ω1 and ω2 are the optimal weights. The sampling frequency is ∆ = 5 minutes.

and the Parkinson ranges at the winning ∆ frequencies were found to be the best choices asexploratory variables in the volatility equation. Results indicate that the Parkinson range shouldbe preferable when analyzing the stocks PETR4, BBDC4, CSNA3, OGXP3, and ITUB4, andthat the Rogers and Satchell range should be used in the case of VALE5, whereas both rangesprovided similar performance in the cases of TNLP4 and USIM5. The the plain version Rt and

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the extension Rωt are the most successful types.

Since the most important feature of the (E)GARCH volatility is its persistence, one way toassess the goodness of the modeling strategy is computing the persistence percentage decrease,that is, the change in the estimated persistence relative to the simple model. The smaller this(negative) quantity the more responsible is the exogenous variable for explaining the conditionalvolatility, decreasing the contribution of the past values. Tables 3 and 4 give, respectively forthe GARCH and the EGARCH models, for all stocks and for some selected frequencies andrealized measures, the estimate of the exogenous variable, the values of the persistence underthe simple and extended models, along with the values of the persistence percentage reduction(PPR). We observe a considerable reduction in the persistence due to the presence of the realizedrange in the volatility equation. The tables also provide the p-value of the likelihood ratio test(LRT) whose null hypothesis is that the simple model (with no exogenous variable) should bepreferable. In table 3, the positive sign of the estimate of θ1 is expected indicating that highervalues for the range are associated with higher levels of the conditional volatility.

Figure 1 plots for some selected fits presented in the previous tables the evolution throughtime of the conditional volatility estimated under the four models. As expected, they are notvery close.

Probably the most difficult step when assessing forecasts performance is the choice of thevolatility proxy, since the true volatility σt is not observed. Recently, the realized volatility hasbeen preferred upon the squares or the absolute values of returns (see, for example, Andersenand Bollerslev (1998)). We use as proxy for σt the 1-minute realized volatility, that is, thesquare root of RVt equation (1), which we denote by σ̃t . The forecast errors are then defined aset = σ̂t− σ̃t , where σ̂t is the forecast under some GARCH-type model.

The out-of-sample one-step-ahead volatility forecasts exercise is carried on as follows. Foreach selected series we separate the initial 400 observations (data from December 3rd, 2009to July 18th, 2011), estimate the (E)GARCH models with and without the realized range, andcompute the one-step-ahead out-of-sample corresponding forecasts. Then we move the datawindow incorporating the next observation and keeping the same sample size of 400. Param-eters are re-estimated. This results in 4 series of 171 out-of-sample one-step-ahead forecastserrors, denoted by ei,t , i = 1,2,3,4, respectively the forecast errors from models GARCH,GARCH+R∗t , EGARCH, and EGARCH+R∗t . The 8 selected model specifications for whichwe present the forecasts results are marked with an s in Table 2.

A simple scale independent measure of performance may be computed using the abso-lute values of the ei,t , t = 1, · · · ,171, see Hyndman(2006). It is the mean absolute percent-age error, defined as mapei = mean( |ei,t |

σ̃t) ∗ 100. We also report a more robust measure of

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Table 3: Some selected GARCH(1,1) and GARCH(1,1)+R∗t fits.Stock (Rt , ∆, R∗t , Cond.Distribution) GARCH(1,1) GARCH(1,1) with R∗t PPR LRT

α1 +β1 α1 +β1 θ1

BBDC4 (RS, 420min, R+ont , t) 0.8421 0.0815∗ -12.7% 0.0003

BBDC4 (RS, 210min, Rωt , t) 0.9646 0.8453 0.1098∗ -12.4% 0.0000

BBDC4 (RS, 210min, R+ont , t) 0.8463 0.0806∗ -12.3% 0.0001

VALE5 (Park, 30min, R+ont , t) 0.8011 0.1513∗ -18.2% 0.0002

VALE5 (Park, 210min, Rωt , t) 0.9792 0.8031 0.1431∗ -18.0% 0.0002

VALE5 (Park, 420min, R+ont , t) 0.8259 0.1238∗ -15.7% 0.0002

ITUB4 (GK1, 420min, R+ont , t) 0.7870 0.1157∗ -18.4% 0.0003

ITUB4 (GK2, 420min, R+ont , t) 0.9642 0.7868 0.1160∗ -18.4% 0.0003

ITUB4 (RS, 420min, R+ont , t) 0.7963 0.1074∗ -17.4% 0.0002

PETR4 (RS, 1sec, Rωt , N) 0.3719 0.4237∗∗ -60.8% 0.0000

PETR4 (Park, 1sec, Rωt , N) 0.9498 0.3772 0.4751∗∗ -60.3% 0.0001

PETR4 (GK1, 1sec, Rωt , N) 0.3799 0.4412∗∗ -60.0% 0.0000

TNLP4 (Park, 10min, Rωt , t) 0.8920 0.1110∗∗ -3.9% 0.0001

TNLP4 (Park, 5min, Rωt , t) 0.9283 0.8975 0.1057∗∗ -3.3% 0.0002

TNLP4 (RS, 30min, Rt , t) 0.9166 0.1061∗∗ -1.3% 0.0001CSNA3 (GK1, 1min, R+on

t , t) 0.8883 0.0897∗ -9.4% 0.0000CSNA3 (GK2, 1min, R+on

t , t) 0.9805 0.8883 0.0898∗ -9.4% 0.0000CSNA3 (RS, 1min, R+on

t , t) 0.8899 0.0836∗ -9.2% 0.0000USIM5 (GK1, 1sec, Rt , N) 0.8175 0.4708∗ -12.3% 0.0000USIM5 (GK1, 1sec, Rscale

t , N) 0.9322 0.8175 0.0887∗ -12.3% 0.0000USIM5 (GK2, 1sec, Rt , N) 0.8168 0.4706∗ -12.4% 0.0000OGXP3 (RS, 210min, Rω

t , t) 0.4477 0.3033∗ -51.9% 0.0000OGXP3 (RS, 210min, Rt , t) 0.9310 0.4668 0.4168∗ -49.9% 0.0000OGXP3 (RS, 210min, Rscale

t , t) 0.4668 0.2935∗ -49.9% 0.0000∗ and ∗∗ denote respectively significance at the 5% and at the 1% level. PPR stands for percentage persistence reduction, and LRT stands for

likelihood ratio test.

discrepancy based on the median, namely, the median absolute percentage error, defined asmdapei =median( |ei,t |

σ̃t)∗100. Table 5 reports the mapei and the mdapei for models i= 1, · · · ,4,

and for the selected stocks. Overall, we observe smaller values for the augmented models, withoccasional disagreement between the two measures. In both cases where the simpler modelseems to be providing forecasts closer to the proxy, the result could be a tie if measures’ vari-ability would have been taken into account.

We now formally test which model specification produces the better one-step-ahead volatil-ity forecasts. To this end we use an statistic suggested in Diebold and Mariano (1995) andmodified in Harvey et al. (1997), which compares the forecasts from two competing models, i

and j. Consider the differences dt = g(ei,t)−g(e j,t), t = 1, · · · ,T , where T is the sample size,here T = 171, and where g(x) typically is the square or the absolute value of x. Christoffersen

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Table 4: Some selected EGARCH(1,1) and EGARCH(1,1)+R∗t fits.Stock (Rt , ∆, R∗t , Cond.Distribution) EGARCH(1,1) EGARCH(1,1) with R∗t PPR LRT

β1 β1 θ1

BBDC4 (Park, 1sec, lnRωt , t) s 0.9414 0.0421∗ -4.21% 0.0332

BBDC4 (GK2, 1sec, lnRt , t) 0.9828 0.9424 0.0417∗∗ -4.11% 0.0238BBDC4 (GK1, 1sec, lnRt , t) 0.9427 0.0413∗∗ -4.08% 0.0239VALE5 (Park, 210min, lnRt , N) 0.8386 0.1141∗ -10.86% 0.0008VALE5 (Park, 60min, lnRt , N) 0.9408 0.8126 0.1451∗ -13.63% 0.0005VALE5 (Park, 105min, lnRscale

t , N) s 0.8428 0.1147∗ -10.42% 0.0009ITUB4 (Park, 1sec, lnRt , N) 0.9726 0.0190∗∗ -1.55% 0.0267ITUB4 (RS, 1sec, lnRt , N) s 0.9879 0.9660 0.0243∗∗ -2.22% 0.0078ITUB4 (RS, 15min, lnRt , N) 0.9578 0.0352∗∗ -3.05% 0.0198PETR4 (Park, 210min, lnRt , N) 0.5776 0.3433∗∗ -36.62% 0.0000PETR4 (Park, 210min, lnRω

t , N) s 0.9114 0.5543 0.3409∗∗ -39.18% 0.0000PETR4 (Park, 210min, R+on

t , N) 0.6699 0.0440∗ -26.50% 0.0002TNLP4 (RS, 420min, lnRt , t) s 0.4282 0.4376∗∗ -55.12% 0.0000TNLP4 (Park, 1sec, lnRt , t) 0.9541 0.9438 0.0344∗∗ -1.08% 0.0161TNLP4 (Park, 1sec, Rω

t , t) 0.9455 0.0089∗∗ -0.90% 0.0122CSNA3 (Park, 1sec, lnR+on

t , t) 0.9463 0.0374∗∗ -4.54% 0.0136CSNA3 (RS, 420min, lnRscale

t , t) s 0.9913 0.9448 0.0527∗∗ -4.69% 0.0011CSNA3 (RS, 1sec, lnR+on

t , t) 0.9567 0.0270∗∗ -3.49% 0.0174USIM5 (Park, 1sec, lnR+on

t , t) s 0.8227 0.1030∗ -13.74% 0.0001USIM5 (RS, 1sec, lnR+on

t , t) 0.9537 0.8429 0.0898∗∗ -11.62% 0.0000USIM5 (RS, 1sec, lnRω

t , t) 0.8271 0.0938∗∗ -13.27% 0.0000OGXP3 (Park, 210min, lnRt , t) s 0.6422 0.2901∗∗ -31.54% 0.0000OGXP3 (RS, 1min, lnRt , t) 0.9380 0.6699 0.2644∗ -28.58% 0.0000OGXP3 (Park, 210min, lnR+on

t , t) 0.7241 0.2114∗∗ -22.80% 0.0000∗ and ∗∗ denote respectively significance at the 5% and at the 1% level. PPR stands for percentage persistence reduction, and LRT stands for

likelihood ratio test.

and Diebold (1996) suggested penalizing differently over and under predictions. However, dueto the uncertainty on the best proxy for σt we just take g(x) as x2. The test statistic is based onthe mean difference d̄ =mean(dt). The null hypothesis is that model j provides better forecasts.Small negative values of d̄ lead to the rejection of the null. Asymptotically the standardized d̄

follows a standard normal distribution, see details in Diebold and Mariano (1995).We carry on three out-of-sample forecasts comparisons. The comparison C1: GARCH+R∗t

versus GARCH, C2: EGARCH+R∗t versus EGARCH, and C3: EGARCH+R∗t versus GARCH+R∗t .Table 6 gives the value of the test statistic and corresponding p-value, for a selection of stocks.The boldface type indicate the rejection of the null hypothesis at the 5% level. Results in the ta-ble are a very good summary of what has occurred for the remaining models specifications. Thatis, typically the GARCH+R∗t and the EGARCH+R∗t models out-perform the simpler GARCH

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Table 5: Mean absolute percentage error (mape) and the median absolute percentage error(mdape) used to compare the GARCH, GARCH+R∗t , EGARCH, and EGARCH+R∗t volatilityforecast.

Stock (Rt , ∆, R∗t , Cond.Distribution) mape mdapeG G+R∗t E E+R∗t G G+R∗t E E+R∗t

BBDC4 (Park, 1sec, lnRωt , t) 18.44 17.89 18.71 18.51 13.12 13.78 16.87 15.08

VALE5 (Park, 105min, lnRscalet , N) 43.82 34.75 38.53 33.53 38.62 28.04 35.06 29.72

ITUB4 (RS, 1sec, lnRt , N) 23.29 21.08 17.70 17.35 17.25 18.02 15.58 15.35PETR4 (Park, 210min, lnRω

t , N) 30.80 24.72 25.18 24.59 24.82 20.45 21.11 20.48TNLP4 (RS, 420min, lnRt , t) 20.31 17.70 20.42 18.23 17.98 15.86 18.49 16.35CSNA3 (RS, 420min, lnRscale

t , t) 14.34 14.44 16.23 16.02 10.77 12.54 14.22 12.83USIM5 (Park, 1sec, lnR+on

t , t) 14.16 16.58 14.04 18.58 11.58 12.70 10.51 14.24OGXP3 (Park, 210min, lnRt , t) 28.72 20.76 22.97 20.85 22.31 17.05 18.11 17.40

Notation in table: Stock (Rt , ∆, R∗t , Cond.Distribution) stands for Asset tick (Realized range, sampling frequency Delta, extended realized

measure R∗t , conditional distribution). G and E stands for models GARCH and EGARCH. The boldface type indicates the model with the

smaller mape or mdape.

Table 6: Diebold and Mariano Test for comparing volatility forecasts performances of [Modeli vs Model j]. Null hypothesis is that model j provides better out-of-sample forecasts. Smallervalues of the test statistic lead to the rejection of the null.

Stock (Rt , ∆, R∗t , Cond.Distribution) C1 C2 C3GARCH+R∗t vs GARCH EGARCH+R∗t vs EGARCH EGARCH+R∗t vs GARCH+R∗t

BBDC4 (Park, 1sec, lnRωt , t) -2.4776 -2.8983 0.1033

(0.0071) (0.0021) (0.5411)

VALE5 (Park, 105min, lnRscalet , N) -6.8324 -3.3370 -0.1828

(0.0000) (0.0005) (0.4276)

ITUB4 (RS, 1sec, lnRt , N) -3.0657 -1.8894 -1.8584(0.0013) (0.0303) (0.0324)

PETR4 (Park, 210min, lnRωt , N) -3.3720 0.7940 -0.5717

(0.0005) (0.7858) (0.2841)

TNLP4 (RS, 420min, lnRt , t) -3.9424 -3.6698 -0.2855(0.0001) (0.0002) (0.3878)

CSNA3 (RS, 420min, lnRscalet , t) -1.8079 -1.6225 -0.0902

(0.0362) (0.0533) (0.4641)

USIM5 (Park, 1sec, lnR+ont , t) 0.2325 0.7854 2.0903

(0.5918) (0.7833) (0.9810)

OGXP3 (Park, 210min, lnRt , t) -3.3857 -1.5215 0.4324(0.0004) (0.0650) (0.6670)

Notation in table: Stock (Rt , ∆, R∗t , Cond.Distribution) stands for Asset tick (Realized range, sampling frequency Delta, extended realized

measure R∗t , conditional distribution). The boldface type indicates rejection of the null at the 5% significance level.

and EGARCH volatility forecasts, whereas there is no clear improvement when moving fromthe GARCH+R∗t to the EGARCH+R∗t .

Finally, Figure 2 shows the persistence evolution during the forecast period for the four

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competing models and for the same stocks of Figure 1. It is clear that the augmented volatilityequation consistently attains remarkable reduction in persistence.

5 Conclusions

In this paper we empirically investigated whether or not the realized range provides additionalinformation to the GARCH and EGARCH model. To this end we considered 4 definitions of therealized range. These measures were computed based on 11 frequencies, as well as 3 possibleways for incorporating the overnight return. This resulted in 176 realized range measures.All, along with their logarithm, were tested as exogenous variables in the conditional varianceequation.

In summary, we found that the realized range is very successful when explaining part of the(E)GARCH conditional volatility. This modeling strategy reduces the persistence and improvesthe out-of-sample forecasts. We were able to reach a conclusion on the best frequencies (1-second, 210-, 420-, 105- and 30-minutes) and best realized range definition to be used, Parkin-son and Rogers and Satchell, at the plain definition and based on a linear combination with theafter hours returns. A possibility not explored here is to obtain the optimal weights of the linearcombination in a joint estimation with the (E)GARCH parameters.

These findings are very promising whenever high-frequency data are not available, since therealized ranges may be computed over daily sampling intervals, and the information required– open, high, low and close daily prices – are usually available. Although only Brazilian datawere used, we think that findings might be extrapolated to other emerging markets due to theirsimilarities.

6 Appendix

References

Aït-Sahalia, Y., Mykland, P. A., and Zhang, L. (2005). How often to sample a continuous-time process

in the presence of market microstructure noise. Review of Financial Studies, 18(2):351–416.

Aït-Sahalia, Y., Mykland, P. A., and Zhang, L. (2011). Ultra high frequency volatility estimation with

dependent microstructure noise. Journal of Econometrics, 160(1):160–175.

Akgiray, V. (1989). Conditional heteroscedasticity in time series of stock returns: Evidence and forecasts.

The Journal of Business, 62(1):55–80.

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Table 7: Summary Statistics for Daily Stock Returns (%)Asset Mean Median Min. Max. Std. Dev. Skewness Kurtosis

BBDC4 0.0244 0.0332 -6.8371 7.7926 1.6663 0.0052 1.2399∗

VALE5 0.0051 0.0439 -8.4009 5.7984 1.7145 -0.2425∗ 1.9782∗

ITUB4 -0.0020 -0.0271 -7.9596 7.5120 1.7379 -0.0919 1.4905∗

PETR4 -0.0740 0.0000 -8.1362 4.9402 1.7851 -0.4160∗ 1.2992∗

TNLP4 -0.0745 -0.0549 -9.4688 12.0500 1.9379 0.2537∗ 4.4382∗

CSNA3 -0.0719 -0.0455 -9.6496 8.5958 2.0771 -0.0447 1.6088∗

USIM5 -0.1090 -0.0922 -6.8066 11.1372 2.3284 0.2951∗ 0.9432∗

OGXP3 0.0055 0.0000 -16.6313 11.1156 2.6769 -0.8132∗ 5.2743∗

Each sample contains 572 daily observations collected from 12/02/09 to 03/23/12. Kurtosis stands for the excess kurtosis. The ∗ indicates

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Figure 1: Evolution through time of the estimated conditional volatility, from July 19th 2011to March 23rd 2012, under the four models across the eight stocks: BBDC4, ITUB4, PETR4,VALE5, CSNA3, OGXP3, TNLP4, and USIM5. The left column compares the GARCH ( )and the GARCH+R∗t (· · · · · · ) models. The right column compares the models EGARCH ( )and EGARCH+R∗t (· · · · · · ).

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Figure 2: Evolution through time of the estimated persistence, from July 19th 2011 to March23rd 2012, under the four models across the eight stocks: BBDC4, ITUB4, PETR4, VALE5,CSNA3, OGXP3, TNLP4, and USIM5. The left column compares the GARCH ( ) and theGARCH+R∗t (· · · · · · ) models. The right column compares the models EGARCH ( ) andEGARCH+R∗t (· · · · · · ).

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